Chapter 22--Fourier Series: Fundamental Period, Frequency, and Angular Frequency

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Period

Picture of a Sine Wave where f(x)=sin(x)

Long long ago, in a high school class called trigonometry, we leaned about periodic functions. A periodic function is a function that repeats itself over and over for infinity. The period of the function is the distance of one iteration that is infinitely repeating.

A signal $f(t)$ is periodic if, for some $T>0$ and all t,

$f(t+T)=f(t)$<ref>Textbook, 22.1</ref>

The picture to the right shows the plot of the standard sine function whose period is $2\pi$. What the plot does not show is that the line keeps extending and repeating the bumps and valleys over the whole x axis, or $(-\infty ,\infty )$. But wait! Can't the period also be $4\pi$ or $6\pi$? In fact it can. Because the graph of sin(x) repeats itself every $2\pi$ units, the period of the function is actually $2\pi n$ where n is any whole number from zero to $\infty$